id stringlengths 14 14 | text stringlengths 9 3.55k | source stringlengths 1 250 |
|---|---|---|
c_593niuqy9dxq | In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced t... | Spectral theorem |
c_ybwpq6wty0lm | In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. | Spectral theorem |
c_mh7xw9yi01u9 | Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. The spectral theorem also provides a canonical decomposition, called the spectral decomposition, of the underlying vector space on which the operator acts. Augustin-Louis Cauchy p... | Spectral theorem |
c_8gvkimp5ffxs | In addition, Cauchy was the first to be systematic about determinants. The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory. This article mainly focuses on the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However... | Spectral theorem |
c_ioedm639o2nw | In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn equipped with the standard inner product. The Gram–Schmidt process takes a finite, line... | Gram-Schmidt theorem |
c_hdy30d1utabq | In mathematics, particularly linear algebra, a zero matrix is a matrix with all its entries being zero. It is alternately denoted by the symbol O {\displaystyle O} . Some examples of zero matrices are 0 1 , 1 = , 0 2 , 2 = , 0 2 , 3 = , {\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix... | Zero tensor |
c_ite0xjgmi05e | 0 K m , n = {\displaystyle 0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}} The zero matrix is the additive identity in K m , n {\displaystyle K_{m,n}} . That is, for all A ∈ K m , n {\displaystyle A\in K_{m,n}}: 0 K m ... | Zero tensor |
c_fs22v29lmm74 | In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring. The zero matrix also represents the linear transformation which sends all vectors to the zero vector. | Zero tensor |
c_dlzxgtvttv3o | In mathematics, particularly linear algebra, a zero matrix or null matrix is a matrix all of whose entries are zero. It also serves as the additive identity of the additive group of m × n {\displaystyle m\times n} matrices, and is denoted by the symbol O {\displaystyle O} or 0 {\displaystyle 0} followed by subscripts c... | Zero matrix |
c_ugub8kbvoi8n | In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V {\displaystyle V} is a basis for V {\displaystyle V} whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an orthonormal basis. | Orthogonal basis |
c_3f8uytnrhca5 | In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V {\displaystyle V} whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R n {\displaystyle... | Complete orthonormal basis |
c_qclk1ay8fgj7 | {\displaystyle V.} Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of R n {\displaystyle \mathbb {R} ^{n}} under dot product. | Complete orthonormal basis |
c_s18335vmf7di | Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process. In functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces. Given a pre-Hilbert space H , {\di... | Complete orthonormal basis |
c_wqaw0w6q2u19 | In this case, the orthonormal basis is sometimes called a Hilbert basis for H . {\displaystyle H.} | Complete orthonormal basis |
c_32g1ybyxdk34 | Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H , {\displaystyle H,} but it may not be the entire space. If we go on to Hilbert spaces, a non-orthonormal set of vectors having the ... | Complete orthonormal basis |
c_y05n80w4q665 | For instance, any square-integrable function on the interval {\displaystyle } can be expressed (almost everywhere) as an infinite sum of Legendre polynomials (an orthonormal basis), but not necessarily as an infinite sum of the monomials x n . {\displaystyle x^{n}.} A different generalisation is to pseudo-inner produc... | Complete orthonormal basis |
c_9zihynex1m7d | In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalizations in the setting of symplectic geometry. A few important generalizations are... | Schur–Horn theorem |
c_cwakvmzmcgw6 | In mathematics, particularly matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular ... | Pascal matrix |
c_v80m0tpc135i | In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix,... | Stieltjes matrix |
c_n649tbllfz8d | In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising the main diagonal and zero or more diagonals on either side. | Bandwidth (linear algebra) |
c_e6x77i693koh | In mathematics, particularly matrix theory, the n×n Lehmer matrix (named after Derrick Henry Lehmer) is the constant symmetric matrix defined by A i j = { i / j , j ≥ i j / i , j < i . {\displaystyle A_{ij}={\begin{cases}i/j,&j\geq i\\j/i,&j | Lehmer matrix |
c_dy1ibtx053ie | In mathematics, particularly measure theory, a 𝜎-ideal, or sigma ideal, of a sigma-algebra (𝜎, read "sigma," means countable in this context) is a subset with certain desirable closure properties. It is a special type of ideal. Its most frequent application is in probability theory.Let ( X , Σ ) {\displaystyle (X,\Si... | Sigma-ideal |
c_6gh55yo5dora | {\textstyle \bigcup _{n\in \mathbb {N} }A_{n}\in N.} Briefly, a sigma-ideal must contain the empty set and contain subsets and countable unions of its elements. | Sigma-ideal |
c_4fuacz16uho5 | The concept of 𝜎-ideal is dual to that of a countably complete (𝜎-) filter. If a measure μ {\displaystyle \mu } is given on ( X , Σ ) , {\displaystyle (X,\Sigma ),} the set of μ {\displaystyle \mu } -negligible sets ( S ∈ Σ {\displaystyle S\in \Sigma } such that μ ( S ) = 0 {\displaystyle \mu (S)=0} ) is a 𝜎-ideal. ... | Sigma-ideal |
c_hb6ruqe0bkpq | {\displaystyle y.} Thus I {\displaystyle I} contains the bottom element, is downward closed, and satisfies a countable analogue of the property of being upwards directed. A 𝜎-ideal of a set X {\displaystyle X} is a 𝜎-ideal of the power set of X . | Sigma-ideal |
c_y0csji8b67lz | {\displaystyle X.} That is, when no 𝜎-algebra is specified, then one simply takes the full power set of the underlying set. For example, the meager subsets of a topological space are those in the 𝜎-ideal generated by the collection of closed subsets with empty interior. | Sigma-ideal |
c_1bb2s6vwkbdv | In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a sc... | Surface integral |
c_0b674tbbi56j | In mathematics, particularly numerical analysis, the Bramble–Hilbert lemma, named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function u {\displaystyle \textstyle u} by a polynomial of order at most m − 1 {\displaystyle \textstyle m-1} in terms of derivatives of u {\displaystyl... | Bramble–Hilbert lemma |
c_hinpuj9jn5qf | Additional assumptions on the domain are needed for the Bramble–Hilbert lemma to hold. Essentially, the boundary of the domain must be "reasonable". For example, domains that have a spike or a slit with zero angle at the tip are excluded. | Bramble–Hilbert lemma |
c_ccezv0jv6277 | Lipschitz domains are reasonable enough, which includes convex domains and domains with continuously differentiable boundary. The main use of the Bramble–Hilbert lemma is to prove bounds on the error of interpolation of function u {\displaystyle \textstyle u} by an operator that preserves polynomials of order up to m −... | Bramble–Hilbert lemma |
c_kqblxla23ez8 | In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the p-adic logarithm. | P-adic exponential function |
c_kgvqqa4i24lo | In mathematics, particularly q-analog theory, the Ramanujan theta function generalizes the form of the Jacobi theta functions, while capturing their general properties. In particular, the Jacobi triple product takes on a particularly elegant form when written in terms of the Ramanujan theta. The function is named after... | Ramanujan theta function |
c_0m8gmv8gnnza | In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, { 2 , 4 , 6 , 8 , 10 } {\displaystyle \{2,4,6,8,10\}} is a finite set with five elements. The number of elements of ... | Tarski finiteness |
c_86jtq3dtf9b9 | A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite: { 1 , 2 , 3 , … } . | Tarski finiteness |
c_yqdwv16sf5ib | {\displaystyle \{1,2,3,\ldots \}.} Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. | Tarski finiteness |
c_7nacc0tqx57h | In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using recursive ordinal notations. | Nonrecursive ordinals |
c_1prqqq3g0t0y | In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions (inverses of collapses), and a homotopy equivalence is a simple homotopy ... | Simple homotopy |
c_jhm4z1u58mcf | In mathematics, particularly the branch called category theory, a 2-group is a groupoid with a way to multiply objects, making it resemble a group. They are part of a larger hierarchy of n-groups. They were introduced by Hoàng Xuân Sính in the late 1960s under the name gr-categories, and they are also known as categori... | 2-group |
c_e8likhfgp65m | In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier. | Fourier cosine series |
c_qp3c18dkw1tt | In mathematics, particularly the study of Lie groups, a Dunkl operator is a certain kind of mathematical operator, involving differential operators but also reflections in an underlying space. Formally, let G be a Coxeter group with reduced root system R and kv an arbitrary "multiplicity" function on R (so ku = kv when... | Dunkl operator |
c_bulik336a7sw | Dunkl operators were introduced by Charles Dunkl (1989). One of Dunkl's major results was that Dunkl operators "commute," that is, they satisfy T i ( T j f ( x ) ) = T j ( T i f ( x ) ) {\displaystyle T_{i}(T_{j}f(x))=T_{j}(T_{i}f(x))} just as partial derivatives do. Thus Dunkl operators represent a meaningful generali... | Dunkl operator |
c_m9b2sg3v1g3o | In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfe... | G-delta space |
c_41wg0hrcnl6u | In mathematics, particularly topology, a comb space is a particular subspace of R 2 {\displaystyle \mathbb {R} ^{2}} that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb space. The deleted comb space is a variation... | Comb space |
c_bvb6tk59pr8i | In mathematics, particularly topology, a cosmic space is any topological space that is a continuous image of some separable metric space. Equivalently (for regular T1 spaces but not in general), a space is cosmic if and only if it has a countable network; namely a countable collection of subsets of the space such that ... | Cosmic space |
c_9ay1gve15zqn | In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology. | Locally normal space |
c_0h3d7j9crpqi | In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas un... | Coordinate map |
c_jjhjk077zab0 | In mathematics, particularly topology, collections of subsets are said to be locally discrete if they look like they have precisely one element from a local point of view. The study of locally discrete collections is worthwhile as Bing's metrization theorem shows. | Locally discrete collection |
c_z7b6krw16fqz | In mathematics, particularly topology, the K-topology is a topology that one can impose on the set of all real numbers which has some interesting properties. Relative to the set of all real numbers carrying the standard topology, the set K = {1/n | n is a positive integer} is not closed since it doesn't contain its (on... | K-topology |
c_shnspv2yq501 | In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automo... | Homeomorphism group |
c_irki5wkimkau | In mathematics, particularly topology, the tube lemma, also called Wallace's theorem, is a useful tool in order to prove that the finite product of compact spaces is compact. | Tube lemma |
c_btnjnzg04il8 | In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure μ {\displaystyle \mu } satisfies Carleman's condition, there is no other measure ν {\displaystyle \nu } having the same moments as μ . {\displaystyle \mu .} The c... | Carleman's condition |
c_ljvjvex7lasl | In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in n-dimensional Euclidean space Rn. Namely, it bounds the volume of the Minkowski sum o... | Milman's reverse Brunn–Minkowski inequality |
c_67pwbzptdhft | In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration (assuming right-associativity), just as tetration is iterated right-associative exponentiation. It is a binary operation defined with two numbers a and b, where a is tetr... | Pentation |
c_l8cbdxxg58ry | In mathematics, perfectoid spaces are adic spaces of special kind, which occur in the study of problems of "mixed characteristic", such as local fields of characteristic zero which have residue fields of characteristic prime p. A perfectoid field is a complete topological field K whose topology is induced by a nondiscr... | Perfectoid space |
c_9v10g6bk141g | In mathematics, persymmetric matrix may refer to: a square matrix which is symmetric with respect to the northeast-to-southwest diagonal; or a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.The first definition is the most common in the recent literature... | Persymmetric matrix |
c_hach0qq8mjii | In mathematics, perturbation theory works typically by expanding unknown quantity in a power series in a small parameter. However, in a perturbation problem beyond all orders, all coefficients of the perturbation expansion vanish and the difference between the function and the constant function 0 cannot be detected by ... | Perturbation problem beyond all orders |
c_8xzsiiivwl5k | This is because the function e − 1 / z {\displaystyle e^{-1/z}} possesses an essential singularity at z = 0 {\displaystyle z=0} in the complex z {\displaystyle z} -plane, and therefore the function is most appropriately modeled by a Laurent series -- a Taylor series has a zero radius of convergence. Thus, if a physical... | Perturbation problem beyond all orders |
c_goy2xrvraaiq | In mathematics, physics and chemistry, a space group is the symmetry group of a repeating pattern in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of the pattern that leave it unchanged. In three dimensions, space groups are classified into 219... | Space groups |
c_sm7o77hfhy7x | In dimensions other than 3, they are sometimes called Bieberbach groups. In crystallography, space groups are also called the crystallographic or Fedorov groups, and represent a description of the symmetry of the crystal. A definitive source regarding 3-dimensional space groups is the International Tables for Crystallo... | Space groups |
c_ftzoytopw5yy | In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized. In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).In digital... | Sinc function |
c_l94ncxxgd2vg | It is used in the concept of reconstructing a continuous bandlimited signal from uniformly spaced samples of that signal. The only difference between the two definitions is in the scaling of the independent variable (the x axis) by a factor of π. In both cases, the value of the function at the removable singularity at ... | Sinc function |
c_xgqlv31fgntx | The sinc function is then analytic everywhere and hence an entire function. The function has also been called the cardinal sine or sine cardinal function. The term sinc was introduced by Philip M. Woodward in his 1952 article "Information theory and inverse probability in telecommunication", in which he said that the f... | Sinc function |
c_rnoue4dwbnnl | In mathematics, physics, and art, moiré patterns (UK: MWAR-ay, US: mwar-AY, French: ) or moiré fringes are large-scale interference patterns that can be produced when a partially opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moiré interference pattern to appear, the two pat... | Moiré effect |
c_2q300x71derd | In television and digital photography, a pattern on an object being photographed can interfere with the shape of the light sensors to generate unwanted artifacts. They are also sometimes created deliberately – in micrometers they are used to amplify the effects of very small movements. In physics, its manifestation is ... | Moiré effect |
c_083601jhlf7s | In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or – as here – simply a vector) is a geometric object that has both a magnitude (or length) and direction. A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "one who ... | Introduction to the mathematics of general relativity |
c_5z8svy9o8vkd | In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a... | Vector direction |
c_dwo8j8syrd5p | It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplicat... | Vector direction |
c_bbvztz9dywfv | Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement... | Vector direction |
c_sz450wyn96rk | In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier transform) of the structure repeat per unit of distance. The SI unit of spatial ... | Spatial frequency |
c_xm3j5ga7tepb | In image-processing applications, spatial frequency is often expressed in units of cycles per millimeter (mm) or equivalently line pairs per mm. In wave propagation, the spatial frequency is also known as wavenumber. Ordinary wavenumber is defined as the reciprocal of wavelength λ {\displaystyle \lambda } and is common... | Spatial frequency |
c_vn7e1jfb71ok | In mathematics, physics, and theoretical computer graphics, tapering is a kind of shape deformation. Just as an affine transformation, such as scaling or shearing, is a first-order model of shape deformation, tapering is a higher order deformation just as twisting and bending. Tapering can be thought of as non-constant... | Tapering (mathematics) |
c_5bdfynmf6aq8 | To create a nonlinear taper, instead of scaling in x and y for all z with constants as in: q = p , {\displaystyle q={\begin{bmatrix}a&0&0\\0&b&0\\0&0&1\end{bmatrix}}p,} let a and b be functions of z so that: q = p . {\displaystyle q={\begin{bmatrix}a(p_{z})&0&0\\0&b(p_{z})&0\\0&0&1\end{bmatrix}}p.} An example of a li... | Tapering (mathematics) |
c_2epvk9j76aku | In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows ho... | Frequency space |
c_t4ly38q8h2tb | Although it is common to refer to the magnitude portion as the frequency response of a signal, the phase portion is required to uniquely define the signal. A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourie... | Frequency space |
c_5jms28emk7sm | The "spectrum" of frequency components is the frequency-domain representation of the signal. The inverse Fourier transform converts the frequency-domain function back to the time-domain function. A spectrum analyzer is a tool commonly used to visualize electronic signals in the frequency domain. | Frequency space |
c_i3vix4m81weg | A frequency-domain representation may describe either a static function or a particular time period of a dynamic function (signal or system). The frequency transform of a dynamic function is performed over a finite time period of that function and assumes the function repeats infinitely outside of that time period. Som... | Frequency space |
c_adn789blwi80 | In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set S ⊂ N {\displaystyle S\subset \mathbb {N} } is called piecewise syndetic if there exists a finite subset G of N {\displaystyle \mathbb {N} } such that for every finite subset F of N {\displaystyle \mathbb {N} } there... | Piecewise syndetic set |
c_vwqu5r432p7h | In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect ... | Planar algebra |
c_lqbco30yuczw | In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Ri... | Plurisubharmonic function |
c_27p9z0nbh37r | In mathematics, point-free geometry is a geometry whose primitive ontological notion is region rather than point. Two axiomatic systems are set out below, one grounded in mereology, the other in mereotopology and known as connection theory. Point-free geometry was first formulated in Whitehead (1919, 1920), not as a th... | Point-free geometry |
c_c8sjjx4go882 | In mathematics, pointless topology, also called point-free topology (or pointfree topology) and locale theory, is an approach to topology that avoids mentioning points, and in which the lattices of open sets are the primitive notions. In this approach it becomes possible to construct topologically interesting spaces fr... | Pointless topology |
c_9xykv3i7pcqd | In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared. | Topology of pointwise convergence |
c_coeg5gnr14qt | In mathematics, poly-Bernoulli numbers, denoted as B n ( k ) {\displaystyle B_{n}^{(k)}} , were defined by M. Kaneko as L i k ( 1 − e − x ) 1 − e − x = ∑ n = 0 ∞ B n ( k ) x n n ! {\displaystyle {Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum _{n=0}^{\infty }B_{n}^{(k)}{x^{n} \over n!}} where Li is the polylogarithm. The B n ( ... | Poly-Bernoulli number |
c_7fs9zc6lh4sq | Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined as follows L i k ( 1 − ( a b ) − x ) b x − a − x c x t = ∑ n = 0 ∞ B n ( k ) ( t ; a , b , c ) x n n ! {\displaystyle {Li_{k}(1-(ab)^{-x}) \over b^{x}-a^{-x}}c^{xt}=\sum _{n=0}^{\infty }B_{n}^{(k)}(t;a,b,c){x^{n} \over n!}} | Poly-Bernoulli number |
c_9ovzirmpfemb | where Li is the polylogarithm. Kaneko also gave two combinatorial formulas: B n ( − k ) = ∑ m = 0 n ( − 1 ) m + n m ! S ( n , m ) ( m + 1 ) k , {\displaystyle B_{n}^{(-k)}=\sum _{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k},} B n ( − k ) = ∑ j = 0 min ( n , k ) ( j ! ) | Poly-Bernoulli number |
c_6xbwoj8buf9m | 2 S ( n + 1 , j + 1 ) S ( k + 1 , j + 1 ) , {\displaystyle B_{n}^{(-k)}=\sum _{j=0}^{\min(n,k)}(j! )^{2}S(n+1,j+1)S(k+1,j+1),} where S ( n , k ) {\displaystyle S(n,k)} is the number of ways to partition a size n {\displaystyle n} set into k {\displaystyle k} non-empty subsets (the Stirling number of the second kind). A... | Poly-Bernoulli number |
c_36ntadww7ffk | Also it is the number of open tours by a biased rook on a board 1 ⋯ 1 ⏟ n 0 ⋯ 0 ⏟ k {\displaystyle \underbrace {1\cdots 1} _{n}\underbrace {0\cdots 0} _{k}} (see A329718 for definition). The Poly-Bernoulli number B k ( − k ) {\displaystyle B_{k}^{(-k)}} satisfies the following asymptotic: B k ( − k ) ∼ ( k ! ) 2 1 k π ... | Poly-Bernoulli number |
c_hx4rwkz6awb1 | {\displaystyle B_{k}^{(-k)}\sim (k! )^{2}{\sqrt {\frac {1}{k\pi (1-\log 2)}}}\left({\frac {1}{\log 2}}\right)^{2k+1},\quad {\text{as }}k\rightarrow \infty .} For a positive integer n and a prime number p, the poly-Bernoulli numbers satisfy B n ( − p ) ≡ 2 n ( mod p ) , {\displaystyle B_{n}^{(-p)}\equiv 2^{n}{\pmod {p}}... | Poly-Bernoulli number |
c_k5r2eszmhe3i | In mathematics, polyad is a concept of category theory introduced by Jean Bénabou in generalising monads. A polyad in a bicategory D is a bicategory morphism Φ from a locally punctual bicategory C to D, Φ: C → D. (A bicategory C is called locally punctual if all hom-categories C(X,Y) consist of one object and one morph... | Polyad |
c_ri9xepz39k6p | In mathematics, polynomial identity testing (PIT) is the problem of efficiently determining whether two multivariate polynomials are identical. More formally, a PIT algorithm is given an arithmetic circuit that computes a polynomial p in a field, and decides whether p is the zero polynomial. Determining the computation... | Polynomial identity testing |
c_tru5p1cjg1d2 | In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: Positive-definite bilinear form Positive-definite function Positive-definite function on a group Positive-definite functional Pos... | Positive definite |
c_ho71x7cdo6h8 | In mathematics, potential flow around a circular cylinder is a classical solution for the flow of an inviscid, incompressible fluid around a cylinder that is transverse to the flow. Far from the cylinder, the flow is unidirectional and uniform. The flow has no vorticity and thus the velocity field is irrotational and c... | Potential flow around a circular cylinder |
c_jx5xastgcral | In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm will produce a number λ {\displaystyle \lambda } , which is the greatest (in absolute value) eigenvalue of A {\displaystyle A} , and a nonzero vector v {\display... | Power method |
c_1qnip05wdm6a | In mathematics, precisely in the theory of functions of several complex variables, a pluriharmonic function is a real valued function which is locally the real part of a holomorphic function of several complex variables. Sometimes such a function is referred to as n-harmonic function, where n ≥ 2 is the dimension of th... | Pluriharmonic function |
c_hhasd5wyex6k | In mathematics, primitive recursive set functions or primitive recursive ordinal functions are analogs of primitive recursive functions, defined for sets or ordinals rather than natural numbers. They were introduced by Jensen & Karp (1971). | Primitive recursive ordinal function |
c_dmu24o01ilzf | In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers R ≥ 0, but in distribution functions.Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mappin... | Probabilistic metric space |
c_7ktvxchdw386 | In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being a... | Progressively measurable |
c_nu8ok399s260 | In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. The simplest case, when the sets are affine spaces, was analyzed by Jo... | Projections onto convex sets |
c_ettz1tcmagmy | For general closed convex sets, the limit point need not be the projection. Classical work on the case of two closed convex sets shows that the rate of convergence of the iterates is linear. There are now extensions that consider cases when there are more than two sets, or when the sets are not convex, or that give fas... | Projections onto convex sets |
c_jdib6urwyacb | Analysis of POCS and related methods attempt to show that the algorithm converges (and if so, find the rate of convergence), and whether it converges to the projection of the original point. These questions are largely known for simple cases, but a topic of active research for the extensions. There are also variants of... | Projections onto convex sets |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.